Mr. Integral

Model a changing world

Differential Equations

Turn rates of change into models of motion, growth and oscillation, then learn how to solve them.

Start learning 7 units · 32 lessons

The course, unit by unit.

Open a unit for its introduction, map and lesson route.

01Equations and Initial ValuesIdentify an ODE, verify solutions and understand what an initial value determines.3 lessons

What can a rate law tell us before we solve it?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Differential Equations and Solution Families
  2. 02Initial Values, Existence and Uniqueness
  3. 03Direction Fields and Solution Curves
02First-Order MethodsSeparate variables, use integrating factors and study nonlinear equilibria.6 lessons

Which structure makes a first-order equation solvable?

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Unit cheat sheet All the key ideas in one reference
  1. 01Separable Equations and Lost Equilibria
  2. 02First-Order Linear Equations
  3. 03Exact Equations and Potential Curves
  4. 04Bernoulli Equations and Substitutions
  5. 05Equilibria, Stability and Logistic Growth
  6. 06Cooling, Mixing and RC Circuits
03Second-Order Linear EquationsBuild homogeneous solutions, handle forcing and explain resonance.6 lessons

How do free motion and forcing combine?

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Unit cheat sheet All the key ideas in one reference
  1. 01Superposition and Fundamental Solutions
  2. 02Characteristic Roots and Free Response
  3. 03Undetermined Coefficients and Resonance
  4. 04Damping and Mechanical Transients
  5. 05Variation of Parameters
  6. 06Frequency Response and RLC Models
04Laplace TransformsConvert initial-value problems into algebra and handle delayed forcing.5 lessons

Can we turn a changing-time problem into algebra?

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Unit cheat sheet All the key ideas in one reference
  1. 01Laplace Transforms and Their Domain
  2. 02Laplace Inversion and Initial-Value Problems
  3. 03Delayed Inputs and Convolution
  4. 04Impulse Inputs and Jump Conditions
  5. 05Transfer Functions, Poles and Response
05Systems and Phase PlanesWrite matrix systems, classify equilibria and linearize nonlinear dynamics.5 lessons

How do coupled variables move together?

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Unit cheat sheet All the key ideas in one reference
  1. 01Linear Systems and Eigenmodes
  2. 02Phase Portraits and Stability
  3. 03Matrix Exponentials and Repeated Modes
  4. 04Forced Systems and Variation of Constants
  5. 05Nonlinear Systems and Local Linearization
06Numerical MethodsApproximate solutions, measure error and recognize stability restrictions.3 lessons

How do we trust a numerical trajectory?

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Unit cheat sheet All the key ideas in one reference
  1. 01Euler’s Method and Local Slopes
  2. 02Midpoint, Heun and Runge–Kutta Methods
  3. 03Numerical Stability, Error and Stiffness
07Series, Boundary Values and CheckpointBuild power-series solutions, resolve periodic inputs into harmonics and understand boundary eigenvalues.4 lessons

What changes when conditions or inputs describe a whole interval?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Power-Series Solutions
  2. 02Fourier Series and Periodic Forcing
  3. 03Boundary-Value Problems and Eigenvalues
  4. 04Differential Equations: Engineering Checkpoint